Integer Programming and Incidence Treedepth
Eduard Eiben, Robert Ganian, Dušan Knop, Sebastian Ordyniak, Michał Pilipczuk, Marcin Wrochna
IInteger Programming and Incidence Treedepth ∗Eduard Eiben † Robert Ganian ‡ Dušan Knop § Sebastian Ordyniak ¶ Michał Pilipczuk ∥ Marcin Wrochna ∗∗ December 2, 2020
Abstract
Recently a strong connection has been shown between the tractability of integer programming (IP)with bounded coefficients on the one side and the structure of its constraint matrix on the other side. Tothat end, integer linear programming is fixed-parameter tractable with respect to the primal (or dual)treedepth of the Gaifman graph of its constraint matrix and the largest coefficient (in absolute value).Motivated by this, Koutecký, Levin, and Onn [ICALP 2018] asked whether it is possible to extend theseresult to a more broader class of integer linear programs. More formally, is integer linear programmingfixed-parameter tractable with respect to the incidence treedepth of its constraint matrix and the largestcoefficient (in absolute value)?We answer this question in negative. In particular, we prove that deciding the feasibility of a systemin the standard form, A x = b , l (cid:54) x (cid:54) u , is NP -hard even when the absolute value of any coefficientin A is 1 and the incidence treedepth of A is 5. Consequently, it is not possible to decide feasibilityin polynomial time even if both the assumed parameters are constant, unless P = NP . Moreover,we complement this intractability result by showing tractability for natural and only slightly morerestrictive settings, namely: (1) treedepth with an additional bound on either the maximum arity ofconstraints or the maximum number of occurrences of variables and (2) the vertex cover number. In this paper we consider the decision version of Integer Linear Program (ILP) in standard form . Here, givena matrix A ∈ Z m × n with m rows (constraints) and n columns and vectors b ∈ Z m and l , u ∈ Z n the taskis to decide whether the set { x ∈ Z n | A x = b , l (cid:54) x (cid:54) u } (SSol) ∗ The manuscript is an extended version of article [88], which appeared in the proceedings of IPCO’19. This work is a partof projects CUTACOMBS, PowAlgDO (M. Wrochna) and TOTAL (M. Pilipczuk) that have received funding from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No.714704, No 714532, and No. 677651). Robert Ganian is supported by the Austrian Science Fund (FWF Project P31336). MarcinWrochna is supported by Foundation for Polish Science (FNP) via the START stipend, and this work was partially done while hewas affiliated with the Institute of Informatics, University of Warsaw, Poland. † Department of Computer Science, Royal Holloway, University of London, UK, [email protected] ‡ Algorithms and Complexity Group, Vienna University of Technology, Austria, [email protected] § Department of Theoretical Computer Science, Czech Technical University in Prague, Czech Republic, [email protected] ¶ School of Computing, University of Leeds, UK, [email protected] ∥ Institute of Informatics, University of Warsaw, Poland, [email protected] ∗∗ University of Oxford, United Kingdom, [email protected] a r X i v : . [ c s . CC ] N ov s non-empty. We are going to study structural properties of the incidence graph of the matrix A . Aninteger program (IP) is a standard IP (SIP) if its set of solutions is described by (SSolSSol), that is, if it is of theform min { f ( x ) | A x = b , l (cid:54) x (cid:54) u , x ∈ Z n } , (SIP)where f : N n → N is the objective function ; in case f is a linear function the above SIP is said to be a linearSIP. Before we go into more details we first review some recent development concerning algorithms forsolving (linear) SIPs in variable dimension with the matrix A admitting a certain decomposition.Let E be a × block matrix, that is, E = (cid:16) A A A A (cid:17) , where A , . . . , A are integral matrices. Wedefine an n -fold 4-block product of E for a positive integer n as the following block matrix E ( n ) = A A A · · · A A A · · · A A · · · ... . . . A · · · A , where is a matrix containing only zeros (of appropriate size). One can ask whether replacing A in thedefinition of the set of feasible solutions (SSolSSol) can give us an algorithmic advantage leading to an efficientalgorithm for solving such SIPs. We call such an SIP an n -fold 4-block IP . We derive two special casesof the n -fold 4-block IP with respect to special cases for the matrix E (see monographs [44, 2121] for moreinformation). If both A and A are void (not present at all), then the result of replacing A with E ( n ) in(SIPSIP) yields the n -fold IP . Similarly, if A and A are void, we obtain the .The first, up to our knowledge, pioneering algorithmic work on n -fold 4-block IPs is due to Hemmeckeet al. [1313]. They gave an algorithm that given n , the × block matrix E , and vectors w , b , l , u finds anintegral vector x with E ( n ) x = b , l (cid:54) x (cid:54) u minimizing wx . The algorithm of Hemmecke et al. [1313]runs in time n g ( r,s, (cid:107) E (cid:107) ∞ ) L , where r is the number of rows of E , s is the number of columns of E , L isthe size of the input, and g : N → N is a computable function. Thus, from the parameterized complexityviewpoint this is an XP algorithm for parameters r, s, (cid:107) E (cid:107) ∞ . This algorithm has been recently improvedby Chen et al. [22] who give better bounds on the function g ; it is worth noting that Chen et al. [22] studyalso the special case where A is a zero matrix and even in that case present an XP algorithm. Since thework of Hemmecke et al. [1313] the question of whether it is possible to improve the algorithm to run in time g (cid:48) ( r, s, (cid:107) E (cid:107) ∞ ) · n O (1) L or not has become a major open question in the area of mathematical programming.Of course, the complexity of the two aforementioned special cases of n -fold 4-block IP are extensivelystudied as well. The first FPT algorithm for the n -fold IPs (for parameters r, s, (cid:107) E (cid:107) ∞ ) is due to Hemmeckeet al. [1414]. Their algorithm has been subsequently improved [1818, 99]. Altmanová et al. [11] implemented thealgorithm of Hemmecke et al. [1414] and improved the polynomial factor (achieving the same running timeas Eisenbrand et al. [99]) the above algorithms (from cubic dependence to n log n ). The best running timeof an algorithm solving n -fold IP is due to Jansen et al. [1616] and runs in nearly linear time in terms of n .Last but not least, there is an FPT algorithm for solving the 2-stage stochastic IP due to Hemmecke andSchultz [1515]. This algorithm is, however, based on a well quasi ordering argument yielding a bound on thesize of the Graver basis for these IPs. Very recently Klein [1717] presented a constructive approach usingSteinitz lemma and give the first explicit (and seemingly optimal) bound on the size of the Graver basis for2-stage (and multistage) IPs. It is worth noting that possible applications of 2-stage stochastic IP are muchless understood than those of its counterpart n -fold IP. That is, an algorithm running in time f ( r, s, (cid:107) E (cid:107) ∞ ) · n O (1) L .
2n the past few years, algorithmic research in this area has been mainly application-driven. Substantialeffort has been taken in order to find the right formalism that is easier to understand and yields algorithmshaving the best possible ratio between their generality and the achieved running time. It turned out thatthe right formalism is connected with variants of the Gaifman graph (see e.g. [55]) of the matrix A (for thedefinitions see the Preliminaries section). Our Contribution.
In this paper we focus on the incidence (Gaifman) graph. We investigate the (negative)effect of the treedepth of the incidence Gaifman graph on tractability of ILP feasibility.
Theorem 1.
Given a matrix A ∈ {− , , } m × n and vectors l , u ∈ Z n ∞ . Deciding whether the set defined by (SSolSSol) is non-empty is NP -hard even if b = and td I ( A ) (cid:54) . We complement Theorem 1Theorem 1 (Section 3Section 3), by showing that (SIPSIP) becomes fixed-parameter tractable parame-terized by (cid:107) A (cid:107) ∞ and either:• treedepth plus min { maxdeg C ( A ) , maxdeg V ( A ) } , where maxdeg C ( A ) is the maximum arity of anyconstraint and maxdeg V ( A ) is the maximum number of constraints any variable occurs in,• the vertex cover number of G I ( A ) . Preliminaries
For integers m < n by [ m : n ] we denote the set { m, m + 1 , . . . , n } and [ n ] is a shorthand for [1 : n ] . Weuse bold face letters for vectors and normal font when referring to their components, that is, x is a vectorand x is its third component. For vectors of vectors we first use superscripts to access the “inner vectors”,that is, x = ( x , . . . , x n ) is a vector of vectors and x is the third vector in this collection. From Matrices to Graphs.
Let A be an m × n integer matrix. The incidence Gaifman graph of A isthe bipartite graph G I = ( R ∪ C, E ) , where R = { r , . . . , r m } contains one vertex for each row of A and C = { c , . . . , c n } contains one vertex for each column of A . There is an edge { r, c } between thevertex r ∈ R and c ∈ C if A ( r, c ) (cid:54) = 0 , that is, if row r contains a nonzero coefficient in column c . The primal Gaifman graph of A is the graph G P = ( C, E ) , where C is the set of columns of A and { c, c (cid:48) } ∈ E whenever there exists a row of A with a nonzero coefficient in both columns c and c (cid:48) . The dual Gaifmangraph of A is the graph G D = ( R, E ) , where R is the set of rows of A and { r, r (cid:48) } ∈ E whenever thereexists a column of A with a nonzero coefficient in both rows r and r (cid:48) . Treedepth.
Undoubtedly, the most celebrated structural parameter for graphs is treewidth, however, inthe case of ILPs bounding treewidth of any of the graphs defined above does not lead to tractability (even ifthe largest coefficient in A is bounded as well see e.g. [1818, Lemma 18]). Treedepth is a structural parameterwhich is useful in the theory of so-called sparse graph classes, see e.g. [2020]. Let G = ( V, E ) be a graph.The treedepth of G , denoted td( G ) , is defined by the following recursive formula: td( G ) = if | V ( G ) | = 1 , v ∈ V ( G ) td( G − v ) if G is connected with | V ( G ) | > , max i ∈ [ k ] td( G i ) if G , . . . , G k are connected components of G .3et A be an m × n integer matrix. The incidence treedepth of A , denoted td I ( A ) , is the treedepth of itsincidence Gaifman graph G I . The dual treedepth of A , denoted td D ( A ) , is the treedepth of its dual Gaifmangraph G D . The primal treedepth is defined similarly.The following two well-known theorems will be used in the proof of Theorem 1Theorem 1. Theorem 2 (Chinese Remainder Theorem).
Let p , . . . , p n be pairwise co-prime integers greater than and let a , . . . , a n be integers such that for all i ∈ [ n ] it holds (cid:54) a i < p i . Then there exists exactly oneinteger x such that1. (cid:54) x < (cid:81) ni =1 p i and2. ∀ i ∈ [ n ] : x ≡ a i mod p i . Theorem 3 (Prime Number Theorem).
Let π ( n ) denote the number of primes in [ n ] , then π ( n ) ∈ Θ( n log n ) . It is worth pointing out that, given a positive integer n encoded in unary, it is possible to the n -th primein polynomial time. Before we proceed to the proof of Theorem 1Theorem 1 we include a brief sketch of its idea. To prove NP -hardness,we will give a polynomial time reduction from 3-SAT which is well known to be NP -complete [1212]. Theproof is inspired by the NP -hardness proof for ILPs given by a set of inequalities, where the primal graph isa star, of Eiben et. al [77]. Proof Idea.
Let ϕ be a 3-CNF formula. We encode an assignment into a variable y . With every variable v i of the formula ϕ we associate a prime number p i . We make y mod p i be the boolean value of the variable v i ; i.e., using auxiliary gadgets we force y mod p i to always be in { , } . Further, if for a clause C ∈ ϕ by (cid:107) C (cid:107) we denote the product of all of the primes associated with the variables occurring in C , then, byChinese Remainder Theorem, there is a single value in [ (cid:107) C (cid:107) ] , associated with the assignment that falsifies C , which we have to forbid for y mod (cid:107) C (cid:107) . We use the box constraints, i.e., the vectors l , u , for an auxiliaryvariable taking the value y mod (cid:107) C (cid:107) to achieve this. For example let ϕ = ( v ∨ ¬ v ∨ v ) and let the primesassociated with the three variables be , , and , respectively. Then we have (cid:107) ( v ∨ ¬ v ∨ v ) (cid:107) = 30 and,since v = v = false and v = true is the only assignment falsifying this clause, we have that isthe forbidden value for y mod 30 . Finally, the (SIPSIP) constructed from ϕ is feasible if and only if there is asatisfying assignment for ϕ .Proof (of Theorem 1Theorem 1). Let ϕ be a 3-CNF formula with n (cid:48) variables v , . . . , v n (cid:48) and m (cid:48) clauses C , . . . , C m (cid:48) (an instance of 3-SAT). Note that we can assume that none of the clauses in ϕ contains a variable along withits negation. We will define an SIP, that is, vectors b , l , u , and a matrix A with O (( n (cid:48) + m (cid:48) ) ) rows andcolumns, whose solution set is non-empty if and only if a satisfying assignment exists for ϕ . Furthermore,we present a decomposition of the incidence graph of the constructed SIP proving that its treedepth is atmost 5. We naturally split the vector x of the SIP into subvectors associated with the sought satisfyingassignment, variables, and clauses of ϕ , that is, we have x = (cid:16) y, x , . . . , x n (cid:48) , z , . . . , z m (cid:48) (cid:17) . Throughoutthe proof p i denotes the i -th prime number. 4 ariable Gadget. We associate the x i = (cid:0) x i , . . . , x ip i (cid:1) part of x with the variable v i and bind theassignment of v i to y . We add the following constraints x i = x i(cid:96) ∀ (cid:96) ∈ [2 : p i ] (1) x i = y + p i (cid:88) (cid:96) =1 x i(cid:96) (2)and box constraints −∞ (cid:54) x i(cid:96) (cid:54) ∞ ∀ (cid:96) ∈ [ p i ] (3) (cid:54) x i (cid:54) (4)to the SIP constructed so far. Claim 1.
For given values of x i and y , one may choose the values of x i(cid:96) for (cid:96) ∈ [ p i ] so that (11) and (22) aresatisfied if and only if x i ≡ y mod p i . Proof of Claim. By (11) we know x i = · · · = x ip i and thus by substitution we get the following equivalentform of (22) x i = y + p i · x i . (5)But this form is equivalent to x i ≡ y mod p i . (cid:121) Note that by (the proof of) the above claim the conditions (11) and (22) essentially replace the large coefficient( p i ) used in the condition (55). This is an efficient trade-off between large coefficients and incidence treedepthwhich we are going to exploit once more when designing the clause gadget.By the above claim we get an immediate correspondence between y and truth assignments for v , . . . , v n (cid:48) .For an integer w and a variable v i we define the following mapping assignment( w, v i ) = true if w ≡ p i false if w ≡ p i undefined otherwise . Notice that (44) implies that the mapping assignment( y, v i ) ∈ { true , false } for i ∈ [ n (cid:48) ] . We straight-forwardly extend the mapping assignment( · , · ) for tuples of variables as follows. For a tuple a of length (cid:96) , thevalue of assignment( w, a ) is (assignment( w, a ) , . . . , assignment( w, a (cid:96) )) and we say that assignment( w, a ) is defined if all of its components are defined. Clause Gadget.
Let C j be a clause with variables v e , v f , v g . We define (cid:107) C j (cid:107) as the product of theprimes associated with the variables occurring in C j , that is, (cid:107) C j (cid:107) = p e · p f · p g . We associate the z j = (cid:16) z j , . . . , z j (cid:107) C j (cid:107) (cid:17) part of x with the clause C j . Let d j be the unique integer in [ (cid:107) C j (cid:107) ] for which assignment( d j , ( v e , v f , v g )) is defined and gives the falsifying assignment for C j . The existence anduniqueness of d j follows directly from the Chinese Remainder Theorem. We add the following constraints z j = z j(cid:96) ∀ (cid:96) ∈ [2 : (cid:107) C j (cid:107) ] (6) z j = y + (cid:88) (cid:54) (cid:96) (cid:54) (cid:107) C j (cid:107) z j(cid:96) (7)5nd box constraints −∞ (cid:54) z j(cid:96) (cid:54) ∞ ∀ (cid:96) ∈ [ (cid:107) C j (cid:107) ] (8) d j + 1 (cid:54) z j (cid:54) (cid:107) C j (cid:107) + d j − (9)to the SIP constructed so far. Claim 2.
Let C j be a clause in ϕ with variables v e , v f , v g . For given values of y and z j such that the value assignment( y, ( v e , v f , v g )) is defined, one may choose the values of z j(cid:96) for (cid:96) ∈ [ (cid:107) C j (cid:107) ] so that (66) , (77) , (88) and (99) are satisfied if and only if assignment( y, ( v e , v f , v g )) satisfies C j . Proof of Claim. Similarly to the proof of the Claim 1Claim 1, (66) and (77) together are equivalent to z j ≡ y mod (cid:107) C j (cid:107) . Finally, by (99) we obtain that z j (cid:54) = d j which holds if and only if assignment( y, ( v e , v f , v g )) satisfies C j . (cid:121) Let A x = be the SIP with constraints (11), (22), (66), and (77) and box constraints l (cid:54) x (cid:54) u given by (33),(44), (88), (99), and −∞ (cid:54) y (cid:54) ∞ . By the Claim 1Claim 1, constraints (11), (22), (33), (44), are equivalent to the assertionthat assignment( y, ( v , . . . , v n (cid:48) )) is defined. Then by the Claim 2Claim 2, constraints (66), (77), (88), (99) are equivalentto checking that every clause in ϕ is satisfied by assignment( y, ( v , . . . , v n (cid:48) )) . This finishes the reductionand the proof of its correctness.In order to finish the proof we have to bound the number of variables and constraints in the presented SIPand to bound the incidence treedepth of A . It follows from the Prime Number Theorem that p i = O ( i log i ) .Hence, the number of rows and columns of A is at most ( n (cid:48) + m (cid:48) ) p n (cid:48) = O (( n (cid:48) + m (cid:48) ) ) . Claim 3.
It holds that td I ( A ) (cid:54) . Proof of Claim. Let G be the incidence graph of the matrix A . It is easy to verify that y is a cut-vertex in G . Observe that each component of G − y is now either a variable gadget for v i with i ∈ [ n (cid:48) ] (we call such acomponent a variable component ) or a clause gadget for C j with j ∈ [ m (cid:48) ] (we call such a component a clausecomponent ). Let G iv be the variable component (of G − y ) containing variables x i and G jc be the clausecomponent containing variables z j . Let t v = max (cid:96) ∈ [ n (cid:48) ] td( G (cid:96)v ) and t c = max (cid:96) ∈ [ m (cid:48) ] td( G (cid:96)c ) . It follows that td( G ) (cid:54) t v , t c ) .Refer to Figure 1Figure 1. Observe that if we delete the variable x i together with the constraint (22) from G iv ,then each component in the resulting graph contains at most two vertices. Each of these componentscontains either• a variable x i(cid:96) and an appropriate constraint (11) (the one containing x i(cid:96) and x i ) for some (cid:96) ∈ [2 : p i ] or• the variable x i .Since treedepth of an edge is 2 and treedepth of the one vertex graph is 1, we have that t v (cid:54) .The bound on t c follows the same lines as for t v , since indeed the two gadgets have the same structure.Now, after deleting z j and (77) in G jc we arrive to a graph with treedepth of all of its components againbounded by two (in fact, none of its components contain more than two vertices). Thus, t v (cid:54) and theclaim follows. (cid:121) The theorem follows by combining Claim 1Claim 1, Claim 2Claim 2, and Claim 3Claim 3. (cid:3) x i = y + (cid:80) p i (cid:96) =1 x i(cid:96) x i x i x i = x i x i = x i · · · x i x ip i · · · Figure 1: The variable gadget for u i of 3-SAT instance together with the global vari-able y . Variables (of the IP) are in circularnodes while equations are in rectangularones. The nodes deleted in the proof ofClaim 3Claim 3 have light gray background. Treedepth and Degree Restrictions
It is worth noting that the proof of Theorem 1Theorem 1 crucially relies onhaving variables as well as constraints which have high degree in the incidence graph. Thus, it is natural toask whether this is necessary or, equivalently, whether bounding the degree of variables, constraints, orboth leads to tractability. It is well known that if a graph G has bounded degree and treedepth, then it is ofbounded size, since indeed the underlying decomposition tree has bounded height and degree and thusbounded number of vertices. Let (SIPSIP) with n variables be given. Let maxdeg C ( A ) denote the maximumarity of a constraint in its constraint matrix A and let maxdeg V ( A ) denote the maximum occurrence of avariable in constraints of A . In other words, maxdeg C ( A ) denotes the maximum number of nonzeros ina row of A and maxdeg V ( A ) denotes the maximum number of nonzeros in a column of A . Now, we getthat ILP can be solved in time f (maxdeg C ( A ) , maxdeg V ( A ) , td I ( A )) L O (1) , where f is some computablefunction and L is the length of the encoding of the given ILP thanks to Lenstra’s algorithm [1919].The above observation can in fact be strengthened—namely, if the arity of all the constraints or thenumber of occurences of all the variables in the given SIP is bounded, then we obtain a bound on eitherprimal or dual treedepth. This is formalized by the following lemma. Lemma 4.
For every (SIPSIP) we have td P ( A ) (cid:54) maxdeg C ( A ) · td I ( A ) and td D ( A ) (cid:54) maxdeg V ( A ) · td I ( A ) . Proof. The proof idea is to investigate the definition of the incidence treedepth of A , which essentiallyboils down to recursively eliminating either a row, or a column, or decomposing a block-decomposablematrix into its blocks. Then, say for the second inequality above, eliminating a column can be replaced byeliminating all the at most maxdeg V ( A ) rows that contain non-zero entries in this column.We now proceed to the proof itself—in particular, we prove only the second inequality, as the first oneis completely symmetric. The proof is uses induction with respect to the total number of rows and columnsof the matrix A . The base of the induction, when A has one row and one column, is trivial, so we proceedto the induction step.Observe that G I ( A ) is disconnected if and only if G D ( A ) is disconnected if and only if A is a block-decomposable matrix. Moreover, the incidence treedepth of A is the maximum incidence treedepth amongthe blocks of A , and the same also holds for the dual treedepth. Hence, in this case we may apply theinduction hypothesis to every block of A and combine the results in a straightforward manner.7ssume then that G I ( A ) is connected. Then td( G I ( A )) = 1 + min v ∈ V ( G I ( A )) td( G I ( A ) − v ) . Let v be the vertex for which the minimum on the right hand side is attained. We consider two cases: either v is a row of A or a column of A .Suppose first that v is a row of A . Then we have td( G D ( A )) (cid:54) G D ( A ) − v ) (cid:54) V ( A ) · td( G I ( A ) − v )= 1 + maxdeg V ( A ) · (td( G I ( A )) − (cid:54) maxdeg V ( A ) · td( G I ( A )) as required, where the second inequality follows from applying the induction assumption to A with therow v removed.Finally, suppose that v is a column of A . Let X be the set of rows of A that contain non-zero entries incolumn v ; then | X | (cid:54) maxdeg V ( A ) and X is non-empty, because G I ( A ) is connected. If we denote by A − v the matrix obtained from A by removing column v , then we have td( G D ( A )) (cid:54) | X | + td( G D ( A ) − X ) (cid:54) maxdeg V ( A ) + td( G D ( A − v )) (cid:54) maxdeg V ( A ) + maxdeg V ( A ) · td( G I ( A − v )) (cid:54) maxdeg V ( A ) · td( G I ( A )) , as required. Here, in the second inequality we used the fact that G D ( A ) − X is a subgraph of G D ( A − v ) ,while in the third inequality we used the induction assumption for the matrix A − v . (cid:3) It follows that if we bound either maxdeg V ( A ) or maxdeg C ( A ) , that is, formally set maxdeg( A ) =min { maxdeg V ( A ) , maxdeg C ( A ) } , then we can use the results of Koutecký et al. [1818] to solve the linearIP with such a solution set in time f (maxdeg( A ) , (cid:107) A (cid:107) ∞ ) · n O (1) · L . Consequently, the use of high-degreeconstraints and variables in the proof of Theorem 1Theorem 1 is unavoidable. Vertex Cover Number
It is natural to ask, whether there are other (more restrictive) structural parame-ters than treedepth that allow for polynomial-time or even fixed-parameter tractability for (SIPSIP). Indeed, onesuch parameter is the (mixed) fracture number of G I ( A ) , which was introduced in [66] and is defined as theminimum integer k such that G I ( A ) has a deletion set D of size at most k ensuring that every componentof G I ( A ) \ D has size at most k . It is easy to see that the treedepth of a graph is upper bounded by twiceits fracture number. It has been shown in [66, Corollary 8] that (SIPSIP) becomes solvable in polynomial-time ifboth the fracture number of G I ( A ) and (cid:107) A (cid:107) ∞ are bounded by a constant. Moreover the question whetherthis result can be improved to fixed-parameter tractability is known to be equivalent to the correspondingand long-standing open questions for 4-block n -fold ILPs [66]. Though we are not able to resolve thisquestion, we can at least show fixed-parameter tractability for a slightly more restrictive parameter thanfracture number, namely, the vertex cover number of G I ( A ) . Towards this result, we need the followingauxiliary corollary, which follows easily from [1111, Theorem 4.1] and shows that (SIPSIP) is fixed-parametertractable parameterized by both the number of rows m in A and (cid:107) A (cid:107) ∞ .8 orollary 5. (SIPSIP) can be solved in time n · O ( m ) m +3 · O ( (cid:107) A (cid:107) ∞ ) m ( m +1) · log( m (cid:107) A (cid:107) ∞ ) , where m and n are the number of rows respectively columns of A . Proof. Eisenbrand and Weismantel recently proved that the corollary holds if all variables in the given(SIPSIP) have a lower bound of , see [1111, Theorem 4.1]. Since one can transform any (SIPSIP) into an (SIPSIP), whereall variables have a lower bound of , by replacing any variable x i , where l i (cid:54) = 0 , with x (cid:48) i + l i , where x (cid:48) i isa new variable with bounds (cid:54) x (cid:48) i (cid:54) u i − l i , and subtracting A ∗ i l i (where A ∗ i denotes the i -th columnof A ) from b , we obtain that [1111, Theorem 4.1] holds for general (SIPSIP). (cid:3) Theorem 6. (SIPSIP) can be solved in time n · O ( k ) k +3 · O ( (cid:107) A (cid:107) ∞ ) k (4 k +2) · log(2 k (cid:107) A (cid:107) ∞ ) , where k is thesize of a minimum vertex cover for G I ( A ) . Proof. Let I be an instance of (SIPSIP) with matrix A . It is well-known, see e.g. [33, Chapter 1], that a minimumvertex cover of an n -vertex graph can be found in time k · O ( kn ) , where k is its size. Hence, we mayassume that we are given a vertex cover C of G I ( A ) of size k . Let O C be the set of all constraints thatcorrespond to vertices in G I ( A ) \ C . Because C is a vertex cover, we obtain that the constraints in O C can only contain the at most k variables in C . Moreover, since we can assume that all rows of A are linearindependent, we obtain that | O C | (cid:54) k . Hence m (cid:54) k and the theorem now follows from Corollary 5Corollary 5. (cid:3) We have shown that, unlike the primal and the dual treedepth, the incidence treedepth of a constraint matrixof (SIPSIP) does not (together with the largest coefficient) provide a way to tractability. This shows that ourcurrent understanding of the structure of the incidence Gaifman graph is not sufficient. Furthermore, it isnot hard to see that the matrix A in our hardness result (cf. Figure 1Figure 1) has topological length (and height ).Topological length is a newly introduced parameter ([1010, Definition 18]) that allows to contract verticesof degree two in the tree witnessing bounded treedepth (i.e., in the tree in whose closure the incidenceGaifman graph emerges as a subgraph). It is worth pointing out that in our reduction we have topologicalheight and constant height while the N -fold -block IP structure implies topological height and theheight of the two levels is an additional parameter. This further stimulates the question of whether an FPT algorithm for N -fold -block IP exists or not. Thus, the effect on tractability of some other “classical” graphparameters shall be investigated.Namely, whether ILP parameterized by the largest coefficient and treewidth and the maximum degreeof the incidence Gaifman graph is in FPT or not. Furthermore, one may also ask about parameterization bythe largest coefficient and the feedback vertex number of the incidence Gaifman graph.
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